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Properties

Label 20.4.43190884988...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{20}\cdot 29^{10}\cdot 179^{4}$
Root discriminant $151.97$
Ramified primes $2, 5, 29, 179$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![594823321, 0, 0, 0, -169747440, 0, -48656055, 0, -8229185, 0, -1247232, 0, -105125, 0, 1015, 0, 805, 0, 50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 50*x^18 + 805*x^16 + 1015*x^14 - 105125*x^12 - 1247232*x^10 - 8229185*x^8 - 48656055*x^6 - 169747440*x^4 + 594823321)
 
gp: K = bnfinit(x^20 + 50*x^18 + 805*x^16 + 1015*x^14 - 105125*x^12 - 1247232*x^10 - 8229185*x^8 - 48656055*x^6 - 169747440*x^4 + 594823321, 1)
 

Normalized defining polynomial

\( x^{20} + 50 x^{18} + 805 x^{16} + 1015 x^{14} - 105125 x^{12} - 1247232 x^{10} - 8229185 x^{8} - 48656055 x^{6} - 169747440 x^{4} + 594823321 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43190884988844472906188100000000000000000000=2^{20}\cdot 5^{20}\cdot 29^{10}\cdot 179^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $151.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 29, 179$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{29} a^{10} - \frac{8}{29} a^{8} - \frac{7}{29} a^{6}$, $\frac{1}{29} a^{11} - \frac{8}{29} a^{9} - \frac{7}{29} a^{7}$, $\frac{1}{841} a^{12} - \frac{8}{841} a^{10} - \frac{413}{841} a^{8} - \frac{9}{29} a^{6} - \frac{1}{29} a^{2}$, $\frac{1}{841} a^{13} - \frac{8}{841} a^{11} - \frac{413}{841} a^{9} - \frac{9}{29} a^{7} - \frac{1}{29} a^{3}$, $\frac{1}{24389} a^{14} - \frac{8}{24389} a^{12} - \frac{413}{24389} a^{10} - \frac{357}{841} a^{8} - \frac{6}{29} a^{6} - \frac{117}{841} a^{4} - \frac{10}{29} a^{2}$, $\frac{1}{24389} a^{15} - \frac{8}{24389} a^{13} - \frac{413}{24389} a^{11} - \frac{357}{841} a^{9} - \frac{6}{29} a^{7} - \frac{117}{841} a^{5} - \frac{10}{29} a^{3}$, $\frac{1}{707281} a^{16} - \frac{8}{707281} a^{14} - \frac{413}{707281} a^{12} - \frac{357}{24389} a^{10} + \frac{197}{841} a^{8} - \frac{6004}{24389} a^{6} - \frac{300}{841} a^{4} - \frac{1}{29} a^{2}$, $\frac{1}{707281} a^{17} - \frac{8}{707281} a^{15} - \frac{413}{707281} a^{13} - \frac{357}{24389} a^{11} + \frac{197}{841} a^{9} - \frac{6004}{24389} a^{7} - \frac{300}{841} a^{5} - \frac{1}{29} a^{3}$, $\frac{1}{5456239467046839208021921} a^{18} - \frac{3636667189286535595}{5456239467046839208021921} a^{16} - \frac{14945419153149674764}{5456239467046839208021921} a^{14} + \frac{21510219366085060432}{188146188518856524414549} a^{12} + \frac{97626246895448289367}{6487799604098500841881} a^{10} + \frac{62962088309083719766230}{188146188518856524414549} a^{8} + \frac{1179475338520101376977}{6487799604098500841881} a^{6} - \frac{103445290029471569610}{223717227727534511789} a^{4} - \frac{3556561873357555696}{7714387163018431441} a^{2} + \frac{20785700472809746}{266013350448911429}$, $\frac{1}{5456239467046839208021921} a^{19} - \frac{3636667189286535595}{5456239467046839208021921} a^{17} - \frac{14945419153149674764}{5456239467046839208021921} a^{15} + \frac{21510219366085060432}{188146188518856524414549} a^{13} + \frac{97626246895448289367}{6487799604098500841881} a^{11} + \frac{62962088309083719766230}{188146188518856524414549} a^{9} + \frac{1179475338520101376977}{6487799604098500841881} a^{7} - \frac{103445290029471569610}{223717227727534511789} a^{5} - \frac{3556561873357555696}{7714387163018431441} a^{3} + \frac{20785700472809746}{266013350448911429} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 115038843562000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.263149228515625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.10.10.16$x^{10} + 10 x + 5$$10$$1$$10$$(C_5^2 : C_8):C_2$$[9/8, 9/8]_{8}^{2}$
5.10.10.16$x^{10} + 10 x + 5$$10$$1$$10$$(C_5^2 : C_8):C_2$$[9/8, 9/8]_{8}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.6.5.1$x^{6} - 29$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$